In a recent paper [1] it was shown that the transition from antiferroelectric to ferroelectric alignment in an antiferroelectric liquid crystal (AFLC) can be homogeneously nucleated only under certain conditions. In particular, the critical electric field was found at which the antiferroelectric configuration loses its stability in an applied field. One of the major simplifications made in Ref. [1] was to neglect molecular rotations in those layers of the AFLC that contained molecules initially aligned in the direction of the applied electric field. In the present work we extend the calculation to include the cooperative motion of all the smectic layers as the transition occurs. It appears that the scenario of the transition from antiferroelectric to ferroelectric state in an AFLC changes drastically as soon as the cooperative motion of all the smectic layers is included. The critical field at which the homogeneous nucleation occurs is now determined mostly by the thickness of the AFLC cell, and not by the molecule-molecule interaction potential. As a result, this field may become much smaller in the model that takes cooperative motion of smectic layers into account than in the model of Ref. [1], which ignores this possibility.
Our starting point is an expression for the
effective Hamiltonian of a cell containing an AFLC.
Figure 1 shows our model. The N smectic layers
lie in the x-z plane and the director is
characterized by the constant angle 
that
it everywhere makes with the y axis and by the variable
azimuthal angle
that it makes relative to the x axis
in the x-z plane
and in layer l. The Hamiltonian is then taken to be
In this expression D is the layer thickness,
k is an elastic constant,
and W(z) is a surface anchoring energy which we take to act only
at the top and bottom surface of the cell, and thus to be of the form
with d the height of the cell.
We assume planar anchoring, so that 
.
The elastic energy terms come from the variation of 
in the
x-z plane. Because each layer is only one molecule thick, there is
no variation of in the y direction within a layer. Instead
there is the interlayer interaction, which is assumed to
favor the 
herringbone structure first considered
by Beresnev [2] and confirmed experimentally by Galerne and
Liebert [3], and by Bahr and Fliegner [4]. Such a structure
has an anti-parallel orientation of adjacent dipoles, as described by the term
with coefficient U. The small chiral
deviation from a perfectly antiparallel orientation is neglected in this
treatment. Finally, there are two terms containing the electric field E,
which
is assumed to be in the z direction. The first of these represents the
effects
of polarization, while the second arises as a consequence of the
dielectric anisotropy, 
. The quantity
is the vacuum permittivity. The
large number of terms in this expression reflects the richness of phases
and complicated dielectric behavior
[5] of the AFLC.